258 XIV INTEGRATION IN LOCALLY COMPACT GROUPS
If G is either compact or discrete, then we have mod(w) = 1 for every
automorphism u of G. For it is clear that w(G) = G and u({e}) = {e}, and we
may apply (14.3.6.2) with A = G and A = {e}, respectively.
We deduce also from (14.3.6,2) that if u and v are two automorphisms of
G, then
(14.3.8) mod(w ° v) = mod(w) mod(v).
(14.3.9) Ifuis any automorphism of the vector space R", then
mod(w) = |det u\.
Let U = (%ij) be the matrix of the automorphism u with respect to the
canonical basis of R". Let Etj denote the n x n matrix which has the element in
the (ij) place equal to 1 and all other elements equal to 0. If / ^j and 1 e R,
put
Then we have the following lemma:
(14.3.9.1) Every invertible n x n matrix U is a product of matrices of the
form Btj(X) and a matrix of the form In + (a~ l)Enn .
Consider invertible matrices of the form
/I 0 0 1
0 {lfB_fc
0 £2,n-h
£i» \
0 0 -
1 £n^h-lin-.h '
«.-*-!..
A A
A e
i: i
I
where /z^0;if/z = 7t-l, then .A" is an arbitrary invertible matrix. The proof
is by induction on h. The matrix BU(X)X is obtained by adding A times the
jth row to the fth row of X. If h = 0 we must have %nn ^ 0; hence if we mul-
tiply on the left successively by the matrices Bin(-%~nl£in) for 1 g i <£ n - 1,
we shall obtain the matrix / + (£nn - \)Enn, which proves the lemma in the
case h = 0. Now suppose that the lemma has been proved for A = 0, 1, ...,
k-\<n, and consider the case h = k. From the expansion for det(JT)
we see Ann 7.4 that there must exist an element £i>n-k ^ 0 for some / such
that n-k-^i^n. Premultiplying X by Bn((\ - £;> ' -*)C«-*) for some index
j ^ i such that n-k^j^n, we see that we may assume that £/,-* = 1-
Multiplying successively by the matrices ^r/(-^r)/J_fc) for r ^j9 we end uposed